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Une approche expérimentale à la théorie algorithmique de la complexité

Hector Zenil 1 
1 SMAC - Systèmes Multi-Agents et Comportements
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
Abstract : In practice, it is a known problem that one cannot compress short strings, shorter, for example, than the length in bits of the compression program which is added to the compressed version of s, making the result (the program producing s) sensitive to the compressor choice and the parameters involved. However, short strings are quite often the kind of data encountered in many practical settings. While compressors' asymptotic behavior guarantees the eventual convergence to K(s) thanks to the invariance theorem, measurements differ considerably in the domain of short strings. We describe a method that combines several theoretical and experimental results to numerically approximate the algorithmic (Kolmogorov-Chaitin) complexity of short bit strings. This is done by an exhaustive execution of abstract machines for which the halting times are known thanks to the Busy Beaver problem. An output frequency distribution is then computed, from which the algorithmic probability is calculated and the algorithmic complexity evaluated by way of the (Levin-Chaitin) coding theorem. The approach we adopt here is different and independent of the machine size (small machines are used only because they allow us to calculate all of them up to a small size) and relies only on the concept of algorithmic probability. The result is a novel approach that we put forward for numerically calculate the complexity of short strings as an alternative to the indirect method using compression algorithms.
Mots-clés : Complexité
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Hector Zenil. Une approche expérimentale à la théorie algorithmique de la complexité. Intelligence artificielle [cs.AI]. Université des Sciences et Technologie de Lille - Lille I, 2011. Français. ⟨NNT : ⟩. ⟨tel-00839374⟩

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